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Beauty of Trajectory

By Hans Christian von Baeyer

I pick up a stone and playfully fling it into Lake Matoaka. The stone rises gracefully through the morning air, tips over, and ends its symmetrical trajectory with a plop. Like countless baseballs, footballs, and basketballs, like lumps of lava hurled out of bubbling volcanoes when Earth was young and drops of water splashed up by oceans till the end of time, my missile traces a mathematical curve through space. We physicists find beauty in the timeless perfection of that motion. It represents a rare glimpse of the absolute in this chaotic world of ours.

Galileo first derived the shape of the trajectory and found it to be the figure that Appolonius of Perga over two millennia ago called a parabola. "Ignoring air resistance, cannon balls move along parabolas," we learn in school. But the truth is more intriguing.

Imagine the stone as a point and the Earth all shriveled up and shrunk down to another point four thousand miles below your feet. This is how Newton, who was born the year Galileo died, imagined it. The relation of the stone to the Earth is exactly the same as that of a comet to the sun, and we know the shape of the comet's path: it is an ellipse.

The true figure of the path of the stone is a skinny ellipse, an oval that is about four thousand miles long and only a few miles wide at its widest, with the Earth's center just inside the lower tip and the stone on the edge of the upper end of the oval. Of course, the stone cannot follow the entire trip, because after just a few seconds it falls into the water -- but that, to the physicist, is an inessential detail. The shape of its path through the air, before it sinks into the lake, is the shape of the upper end of that almost unimaginably skinny ellipse.

The stone traces out before my eyes the trajectory of Comet Halley -- but not the part we see when Halley races in hot fury around the sun -- no, it imitates the other part that we never see, when Halley almost coasts to a stop thirty-eight years later and hundreds of millions of miles from the Sun and starts on its return journey, a thirty-eight year fall toward the Sun. I have always wanted to be there when the great comet, far out in the dark cold of outer space, moving almost imperceptibly slowly, comes to the apex of its odyssey and begins its long haul back home to the warmth of the Sun. But I don't need to go that far away. A stone tossed over Lake Matoaka mimics precisely what I would see out there beyond the orbit of Pluto. It is the business of physics to find unity in the diversity of natural phenomena --and to discover analogies between the inaccessible realms of the universe and the immediate world of human experience.

Hans Christian von Baeyer is Chancellor Professor of Physics College of William and Mary, Williamsburg, VA.